Stabilities for Nonisentropic Euler-Poisson Equations

Abstract

We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces.

1. Introduction

The compressible nonisentropic Euler or Euler-Poisson system for fluids can be written as where is the frictional damping constant and is a constant related to the unit ball in . , and where is the volume of the unit ball in . As usual, , , and are the density, the velocity, and the entropy, respectively. is the pressure function, for which the constants and .

When , the system is self-attractive. The system (1) is the Newtonian description of gaseous stars [1]. When , the system comprises the Euler-Poisson equations with repulsive forces and can be used as a semiconductor model [2, 3]. When , the system comprises the compressible Euler equations and can be applied as a classical model in fluid mechanics [3]. For more classical and recent results in these systems, readers can refer to [1, 4–10].

It is well known that the solution for the Poisson equation (1)₄ can be written as where is the Green’s function for the Poisson equation in the -dimensional spaces defined by

Notation. In the following discussion, classical solutions are solutions with compact support for each fixed time . We also denote the total mass by , where where .

Lastly, we will denote

2. Lemmas

In this section, we establish some lemmas for the proof of the main results. The following lemma will be used to derive the energy functional for ; namely, is conserved in time if the system (1) is not damped.

Lemma 1. For the classical solution of system (1), we have where is defined by (1)₅.

Proof. We have Note that, by Divergence Theorem, Thus,

Next, the results of the following two lemmas will be used in the derivations of the energy functionals for both and . It will be shown that in Section 3 the energy functional for is which is conversed in time if the system (1) is not damped.

Lemma 2. For the classical solution of system (1), we have where is defined by (1)₅ and is the solution of (1)₄.

Proof. We have One can check a detail proof of the following equality in the Appendix: Thus, by (16) and (17).
Thus, the proof is complete.

Lemma 3. For the classical solution of system (1), we have where is the solution of (1)₄.

Proof. We have Thus, the first equality in (19) holds.
Next, Thus, the second equality in (19) holds.

The lemma below is crucial to obtaining the energy functional for . Comparing the left hand sides of (8) and (22), we note that the left hand side of (22) (given in the next lemma), which contains the term , is nontrivial to be found.

Lemma 4. For the classical solution of system (1) with , we have where is defined by (1)₅.

Proof. Note that Thus, The proof is complete.

3. Main Results

In this section, we find out the energy functionals for the system (1) in the case of (Proposition 5) and (Proposition 6). Moreover, we establish the stabilities results (Proposition 8) and a blowup result (Proposition 9) for system (1).

Proposition 5. For the classical solution of system (1) with , let Then, where is the devertive of with respect to .
Thus, is a decreasing function and is conserved if the system is not damped.

Proof. By Lemma 1, By Lemma 2, By Lemma 3, Thus, the proof is complete.

Proposition 6. For the classical solution of system (1) with , let Then, where is the derivative of with respect to .
Thus, is a decreasing function and is conserved if the system is not damped.

Proof. By Lemma 2, By Lemma 3, By Lemma 4, Thus, the proof is complete.

Proposition 7. Let We have where , , is a classical solution of system (1), and is defined by (5).

Proof. We split the last term of the above equality into three parts.
Firstly, Secondly, Thirdly, where is the gradient operator with respect to the spatial variable .
Note that For , Thus, The result for is established.
For , Thus, The results for are also established.

Now, we are ready to present the stability results.

Proposition 8. Considering the classical solutions of system (1), we have the following.
Case 1. For , , or , , and , we have
Case 2. For , , , and , we have
Case 3. For , , , and , we have

Proof. First of all, by definitions (35), (6), and (5), we always have
Case 1. By Propositions 5 and 7, we have Thus, In view of inequality (49), we have Thus,
Case 2. By Propositions 5 and 7, we have Note that is a positive function for by (3) and (4). Thus, It follows that
Case 3. By Proposition 7, we have It follows that

Finally, we can give the blowup result.

Proposition 9. If , , , and , then the classical solutions of (1) blow up in finite time.

Proof.    
Case 1 (). As before, we have, from Propositions 5 and 7, that It follows that Suppose the solutions exist globally; then for sufficient large , we see that is negative as the leading coefficient of the right hand side of (60) is negative. However, is nonnegative by definition (35). This is a contradiction. As a result, the solutions blow up in finite time.
Case 2 (). Now (59) becomes It follows by multipying an integral factor on both sides and taking integration that for some constants and . Note that this implies that is negative for sufficient large as and . Therefore, the solutions blow up in finite time.

Appendix

We here complement the proof of Lemma 2 by proving the equality (17); namely,

Proof. Firstly, by divergence theorem, Secondly, by definitions of the operations, Thus, Thus, equality (A.1) is established.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research paper is partially supported by Grant (MIT/SRG02/14-15) from the Department of Mathematics and Information Technology of the Hong Kong Institute of Education.